Who is Éomer in the movie?

Éomer is a relative to king Théoden, brother to Éowyn. He mistrusts Gríma Wormtongue but he has no prove against the Worm.

Who is Éomer in the books?
Éomer is the son of Théoden‘s sister Théodwyn and Eomund, a Marshal of the Riddermark. Eomund died in a fight with a band of Orcs when Éomer was 11 years old. His mother Théodwyn got ill and passed away shortly after her husband’s death. Thus Éomer and Éowyn where raised by their uncle in Edoras.

Changes in The Two Towers

Théodred and the fords of the Isen
The first time we meet Éomer in the movies, is when he finds his cousin Théodred at the fords of the Isen. This scene is totally made up, the cousins couldn’t meet at that moment, since Éomer was the Marshal of the East-mark, while Théodred was in command of the West-mark.
In the books Éomer was never banished from Rohan. He was imprisoned though, after a similar conflict with Gríma and Théoden but this didn’t happen until after his meeting with Aragorn, Gimli and Legolas.

The meeting with Aragorn, Gimli and Legolas
In the books, this is the first time we see Éomer. The meeting is very similar to the one in the movies, although -as said before- in the books Éomer wasn’t banished, but rode out to protect his country from the band of Orcs against the will of Théoden and Gríma. He then meets Aragorn, Gimli and Legolas and agrees to let them continue their search for the hobbits, although by law only the king can decide who can pass the lands. He lends them the two horses, Arod and Hasufel. Unless Aragorn proves that he is trustworthy by returning the horses later and coming to the king’s court to ask for permission to travel through Rohan, he might forfeit his life by this action. Indeed Gríma grabs the opportunity with both hands to order Éomer’s imprisonment on his return to Meduseld. When Aragorn and the others finally arrive in Meduseld and the king is no longer under the influence of Gríma, Éomer is freed on advice of Gandalf and Háma, and rides out with Théoden King.

Helm’s deep
Éomer leads the army that attacks the Orcs from behind in the movie, while in the books, Gandalf brings a different character named Erkenbrand, who is the lord of Helms Deep, with his troops. Éomer and Aragorn are leading the defence of the Hornburg together. During the fight, Éomer gets separated from the others, and together with Gimli and a number of his men he has to retreat to the Glittering Caves (of which we get a glimpse in the movie as the caves where the women and children hide). After the battle, they return safely to their friends.

Bookie details
– “Then one rode forward, a tall man, taller than all the rest; from his helm as a crest a white horsetail flowed.” The white horsetail can be clearly seen at the first meeting with the three hunters.